arXiv subjects

James B. Simon

Publications and source records attributed to James B. Simon.

At least 19 recordsLinked to original sources

arXiv2026

Saddle-To-Saddle Dynamics in Deep ReLU Networks: Low-Rank Bias in the First Saddle Escape

When a deep ReLU network is initialized with small weights, gradient descent (GD) is at first dominated by the saddle at the origin in parameter space. We study the so-called escape directions along which GD leaves the origin, which play a similar role as the eigenvectors of the Hessian for strict saddles. We show that the optimal escape direction features a low-rank bias in its deeper layers: the first singular value of the $\ell$-th layer weight matrix is at least $\ell^{\frac{1}{4}}$ larger than any other singular value. We also prove a number of related results about these escape directions. We suggest that deep ReLU networks exhibit saddle-to-saddle dynamics, with GD visiting a sequence of saddles with increasing bottleneck rank (Jacot, 2023).

cs.LG↗

arXiv2026

Predicting kernel regression learning curves from only raw data statistics

We study kernel regression with common rotation-invariant kernels on real datasets including CIFAR-5m, SVHN, and ImageNet. We give a theoretical framework that predicts learning curves (test risk vs. sample size) from only two measurements: the empirical data covariance matrix and an empirical polynomial decomposition of the target function $f_*$. The key new idea is an analytical approximation of a kernel's eigenvalues and eigenfunctions with respect to an anisotropic data distribution. The eigenfunctions resemble Hermite polynomials of the data, so we call this approximation the Hermite eigenstructure ansatz (HEA). We prove the HEA for Gaussian data, but we find that real image data is often "Gaussian enough" for the HEA to hold well in practice, enabling us to predict learning curves by applying prior results relating kernel eigenstructure to test risk. Extending beyond kernel regression, we empirically find that MLPs in the feature-learning regime learn Hermite polynomials in the order predicted by the HEA. Our HEA framework is a proof of concept that an end-to-end theory of learning which maps dataset structure all the way to model performance is possible for nontrivial learning algorithms on real datasets.

cs.LG↗

arXiv2025

Alternating Gradient Flows: A Theory of Feature Learning in Two-layer Neural Networks

What features neural networks learn, and how, remains an open question. In this paper, we introduce Alternating Gradient Flows (AGF), an algorithmic framework that describes the dynamics of feature learning in two-layer networks trained from small initialization. Prior works have shown that gradient flow in this regime exhibits a staircase-like loss curve, alternating between plateaus where neurons slowly align to useful directions and sharp drops where neurons rapidly grow in norm. AGF approximates this behavior as an alternating two-step process: maximizing a utility function over dormant neurons and minimizing a cost function over active ones. AGF begins with all neurons dormant. At each iteration, a dormant neuron activates, triggering the acquisition of a feature and a drop in the loss. AGF quantifies the order, timing, and magnitude of these drops, matching experiments across several commonly studied architectures. We show that AGF unifies and extends existing saddle-to-saddle analyses in fully connected linear networks and attention-only linear transformers, where the learned features are singular modes and principal components, respectively. In diagonal linear networks, we prove AGF converges to gradient flow in the limit of vanishing initialization. Applying AGF to quadratic networks trained to perform modular addition, we give the first complete characterization of the training dynamics, revealing that networks learn Fourier features in decreasing order of coefficient magnitude. Altogether, AGF offers a promising step towards understanding feature learning in neural networks.

cs.LG↗

arXiv2025

Closed-Form Training Dynamics Reveal Learned Features and Linear Structure in Word2Vec-like Models

Self-supervised word embedding algorithms such as word2vec provide a minimal setting for studying representation learning in language modeling. We examine the quartic Taylor approximation of the word2vec loss around the origin, and we show that both the resulting training dynamics and the final performance on downstream tasks are empirically very similar to those of word2vec. Our main contribution is to analytically solve for both the gradient flow training dynamics and the final word embeddings in terms of only the corpus statistics and training hyperparameters. The solutions reveal that these models learn orthogonal linear subspaces one at a time, each one incrementing the effective rank of the embeddings until model capacity is saturated. Training on Wikipedia, we find that each of the top linear subspaces represents an interpretable topic-level concept. Finally, we apply our theory to describe how linear representations of more abstract semantic concepts emerge during training; these can be used to complete analogies via vector addition.

cs.LG↗

arXiv2025

The Features at Convergence Theorem: a first-principles alternative to the Neural Feature Ansatz for how networks learn representations

It is a central challenge in deep learning to understand how neural networks learn representations. A leading approach is the Neural Feature Ansatz (NFA) (Radhakrishnan et al. 2024), a conjectured mechanism for how feature learning occurs. Although the NFA is empirically validated, it is an educated guess and lacks a theoretical basis, and thus it is unclear when it might fail, and how to improve it. In this paper, we take a first-principles approach to understanding why this observation holds, and when it does not. We use first-order optimality conditions to derive the Features at Convergence Theorem (FACT), an alternative to the NFA that (a) obtains greater agreement with learned features at convergence, (b) explains why the NFA holds in most settings, and (c) captures essential feature learning phenomena in neural networks such as grokking behavior in modular arithmetic and phase transitions in learning sparse parities, similarly to the NFA. Thus, our results unify theoretical first-order optimality analyses of neural networks with the empirically-driven NFA literature, and provide a principled alternative that provably and empirically holds at convergence.

cs.LG↗

arXiv2025

The Optimization Landscape of SGD Across the Feature Learning Strength

We consider neural networks (NNs) where the final layer is down-scaled by a fixed hyperparameter $γ$. Recent work has identified $γ$ as controlling the strength of feature learning. As $γ$ increases, network evolution changes from "lazy" kernel dynamics to "rich" feature-learning dynamics, with a host of associated benefits including improved performance on common tasks. In this work, we conduct a thorough empirical investigation of the effect of scaling $γ$ across a variety of models and datasets in the online training setting. We first examine the interaction of $γ$ with the learning rate $η$, identifying several scaling regimes in the $γ$-$η$ plane which we explain theoretically using a simple model. We find that the optimal learning rate $η^*$ scales non-trivially with $γ$. In particular, $η^* \propto γ^2$ when $γ\ll 1$ and $η^* \propto γ^{2/L}$ when $γ\gg 1$ for a feed-forward network of depth $L$. Using this optimal learning rate scaling, we proceed with an empirical study of the under-explored "ultra-rich" $γ\gg 1$ regime. We find that networks in this regime display characteristic loss curves, starting with a long plateau followed by a drop-off, sometimes followed by one or more additional staircase steps. We find networks of different large $γ$ values optimize along similar trajectories up to a reparameterization of time. We further find that optimal online performance is often found at large $γ$ and could be missed if this hyperparameter is not tuned. Our findings indicate that analytical study of the large-$γ$ limit may yield useful insights into the dynamics of representation learning in performant models.

cs.LG↗

arXiv2024

Benign, Tempered, or Catastrophic: A Taxonomy of Overfitting

The practical success of overparameterized neural networks has motivated the recent scientific study of interpolating methods, which perfectly fit their training data. Certain interpolating methods, including neural networks, can fit noisy training data without catastrophically bad test performance, in defiance of standard intuitions from statistical learning theory. Aiming to explain this, a body of recent work has studied benign overfitting, a phenomenon where some interpolating methods approach Bayes optimality, even in the presence of noise. In this work we argue that while benign overfitting has been instructive and fruitful to study, many real interpolating methods like neural networks do not fit benignly: modest noise in the training set causes nonzero (but non-infinite) excess risk at test time, implying these models are neither benign nor catastrophic but rather fall in an intermediate regime. We call this intermediate regime tempered overfitting, and we initiate its systematic study. We first explore this phenomenon in the context of kernel (ridge) regression (KR) by obtaining conditions on the ridge parameter and kernel eigenspectrum under which KR exhibits each of the three behaviors. We find that kernels with powerlaw spectra, including Laplace kernels and ReLU neural tangent kernels, exhibit tempered overfitting. We then empirically study deep neural networks through the lens of our taxonomy, and find that those trained to interpolation are tempered, while those stopped early are benign. We hope our work leads to a more refined understanding of overfitting in modern learning.

cs.LG↗

arXiv2024

More is Better in Modern Machine Learning: when Infinite Overparameterization is Optimal and Overfitting is Obligatory

In our era of enormous neural networks, empirical progress has been driven by the philosophy that more is better. Recent deep learning practice has found repeatedly that larger model size, more data, and more computation (resulting in lower training loss) improves performance. In this paper, we give theoretical backing to these empirical observations by showing that these three properties hold in random feature (RF) regression, a class of models equivalent to shallow networks with only the last layer trained. Concretely, we first show that the test risk of RF regression decreases monotonically with both the number of features and the number of samples, provided the ridge penalty is tuned optimally. In particular, this implies that infinite width RF architectures are preferable to those of any finite width. We then proceed to demonstrate that, for a large class of tasks characterized by powerlaw eigenstructure, training to near-zero training loss is obligatory: near-optimal performance can only be achieved when the training error is much smaller than the test error. Grounding our theory in real-world data, we find empirically that standard computer vision tasks with convolutional neural tangent kernels clearly fall into this class. Taken together, our results tell a simple, testable story of the benefits of overparameterization, overfitting, and more data in random feature models.

cs.LG↗

arXiv2024

A Spectral Condition for Feature Learning

The push to train ever larger neural networks has motivated the study of initialization and training at large network width. A key challenge is to scale training so that a network's internal representations evolve nontrivially at all widths, a process known as feature learning. Here, we show that feature learning is achieved by scaling the spectral norm of weight matrices and their updates like $\sqrt{\texttt{fan-out}/\texttt{fan-in}}$, in contrast to widely used but heuristic scalings based on Frobenius norm and entry size. Our spectral scaling analysis also leads to an elementary derivation of \emph{maximal update parametrization}. All in all, we aim to provide the reader with a solid conceptual understanding of feature learning in neural networks.

cs.LG↗

arXiv2024

An Agnostic View on the Cost of Overfitting in (Kernel) Ridge Regression

We study the cost of overfitting in noisy kernel ridge regression (KRR), which we define as the ratio between the test error of the interpolating ridgeless model and the test error of the optimally-tuned model. We take an "agnostic" view in the following sense: we consider the cost as a function of sample size for any target function, even if the sample size is not large enough for consistency or the target is outside the RKHS. We analyze the cost of overfitting under a Gaussian universality ansatz using recently derived (non-rigorous) risk estimates in terms of the task eigenstructure. Our analysis provides a more refined characterization of benign, tempered and catastrophic overfitting (cf. Mallinar et al. 2022).

stat.ML↗

arXiv2024

Les Houches Lectures on Deep Learning at Large & Infinite Width

These lectures, presented at the 2022 Les Houches Summer School on Statistical Physics and Machine Learning, focus on the infinite-width limit and large-width regime of deep neural networks. Topics covered include various statistical and dynamical properties of these networks. In particular, the lecturers discuss properties of random deep neural networks; connections between trained deep neural networks, linear models, kernels, and Gaussian processes that arise in the infinite-width limit; and perturbative and non-perturbative treatments of large but finite-width networks, at initialization and after training.

stat.ML↗

arXiv2023

The Eigenlearning Framework: A Conservation Law Perspective on Kernel Regression and Wide Neural Networks

We derive simple closed-form estimates for the test risk and other generalization metrics of kernel ridge regression (KRR). Relative to prior work, our derivations are greatly simplified and our final expressions are more readily interpreted. These improvements are enabled by our identification of a sharp conservation law which limits the ability of KRR to learn any orthonormal basis of functions. Test risk and other objects of interest are expressed transparently in terms of our conserved quantity evaluated in the kernel eigenbasis. We use our improved framework to: i) provide a theoretical explanation for the "deep bootstrap" of Nakkiran et al (2020), ii) generalize a previous result regarding the hardness of the classic parity problem, iii) fashion a theoretical tool for the study of adversarial robustness, and iv) draw a tight analogy between KRR and a well-studied system in statistical physics.

cs.LG↗

arXiv2023

Tune As You Scale: Hyperparameter Optimization For Compute Efficient Training

Hyperparameter tuning of deep learning models can lead to order-of-magnitude performance gains for the same amount of compute. Despite this, systematic tuning is uncommon, particularly for large models, which are expensive to evaluate and tend to have many hyperparameters, necessitating difficult judgment calls about tradeoffs, budgets, and search bounds. To address these issues and propose a practical method for robustly tuning large models, we present Cost-Aware Pareto Region Bayesian Search (CARBS), a Bayesian optimization algorithm that performs local search around the performance-cost Pareto frontier. CARBS does well even in unbounded search spaces with many hyperparameters, learns scaling relationships so that it can tune models even as they are scaled up, and automates much of the "black magic" of tuning. Among our results, we effectively solve the entire ProcGen benchmark just by tuning a simple baseline (PPO, as provided in the original ProcGen paper). We also reproduce the model size vs. training tokens scaling result from the Chinchilla project (Hoffmann et al. 2022), while simultaneously discovering scaling laws for every other hyperparameter, via an easy automated process that uses significantly less compute and is applicable to any deep learning problem (not just language models).

cs.LG↗

arXiv2023

On the Stepwise Nature of Self-Supervised Learning

We present a simple picture of the training process of joint embedding self-supervised learning methods. We find that these methods learn their high-dimensional embeddings one dimension at a time in a sequence of discrete, well-separated steps. We arrive at this conclusion via the study of a linearized model of Barlow Twins applicable to the case in which the trained network is infinitely wide. We solve the training dynamics of this model from small initialization, finding that the model learns the top eigenmodes of a certain contrastive kernel in a stepwise fashion, and obtain a closed-form expression for the final learned representations. Remarkably, we then see the same stepwise learning phenomenon when training deep ResNets using the Barlow Twins, SimCLR, and VICReg losses. Our theory suggests that, just as kernel regression can be thought of as a model of supervised learning, kernel PCA may serve as a useful model of self-supervised learning.

cs.LG↗

arXiv2023

SGD with a Constant Large Learning Rate Can Converge to Local Maxima

Previous works on stochastic gradient descent (SGD) often focus on its success. In this work, we construct worst-case optimization problems illustrating that, when not in the regimes that the previous works often assume, SGD can exhibit many strange and potentially undesirable behaviors. Specifically, we construct landscapes and data distributions such that (1) SGD converges to local maxima, (2) SGD escapes saddle points arbitrarily slowly, (3) SGD prefers sharp minima over flat ones, and (4) AMSGrad converges to local maxima. We also realize results in a minimal neural network-like example. Our results highlight the importance of simultaneously analyzing the minibatch sampling, discrete-time updates rules, and realistic landscapes to understand the role of SGD in deep learning.

cs.LG↗

arXiv2022

Avalon: A Benchmark for RL Generalization Using Procedurally Generated Worlds

Despite impressive successes, deep reinforcement learning (RL) systems still fall short of human performance on generalization to new tasks and environments that differ from their training. As a benchmark tailored for studying RL generalization, we introduce Avalon, a set of tasks in which embodied agents in highly diverse procedural 3D worlds must survive by navigating terrain, hunting or gathering food, and avoiding hazards. Avalon is unique among existing RL benchmarks in that the reward function, world dynamics, and action space are the same for every task, with tasks differentiated solely by altering the environment; its 20 tasks, ranging in complexity from eat and throw to hunt and navigate, each create worlds in which the agent must perform specific skills in order to survive. This setup enables investigations of generalization within tasks, between tasks, and to compositional tasks that require combining skills learned from previous tasks. Avalon includes a highly efficient simulator, a library of baselines, and a benchmark with scoring metrics evaluated against hundreds of hours of human performance, all of which are open-source and publicly available. We find that standard RL baselines make progress on most tasks but are still far from human performance, suggesting Avalon is challenging enough to advance the quest for generalizable RL.

cs.AI↗

arXiv2022

On Kernel Regression with Data-Dependent Kernels

The primary hyperparameter in kernel regression (KR) is the choice of kernel. In most theoretical studies of KR, one assumes the kernel is fixed before seeing the training data. Under this assumption, it is known that the optimal kernel is equal to the prior covariance of the target function. In this note, we consider KR in which the kernel may be updated after seeing the training data. We point out that an analogous choice of kernel using the posterior of the target function is optimal in this setting. Connections to the view of deep neural networks as data-dependent kernel learners are discussed.

cs.LG↗

arXiv2022

Reverse Engineering the Neural Tangent Kernel

The development of methods to guide the design of neural networks is an important open challenge for deep learning theory. As a paradigm for principled neural architecture design, we propose the translation of high-performing kernels, which are better-understood and amenable to first-principles design, into equivalent network architectures, which have superior efficiency, flexibility, and feature learning. To this end, we constructively prove that, with just an appropriate choice of activation function, any positive-semidefinite dot-product kernel can be realized as either the NNGP or neural tangent kernel of a fully-connected neural network with only one hidden layer. We verify our construction numerically and demonstrate its utility as a design tool for finite fully-connected networks in several experiments.

cs.LG↗